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                                    2.6 地球参考椭球与地球重力场
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                                    2023-10-03, 666 words, 3 min read
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                                        <h2 id="一-地球参考椭球">一、地球参考椭球</h2>
<h4 id="1-圆球体">1、圆球体</h4>
<p>1、方程：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2+y^2+z^2=R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p>
<p>2、适用范围： 对于一般的工程应用</p>
<h4 id="2-旋转椭球体">2、旋转椭球体</h4>
<p>1、方程：<img src="http://cos.pansis.site/202309281508677.png/abc123" alt="image-20230928150807570" style="zoom:50%;" /></p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">R_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为长半轴即地球赤道半径； <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>R</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">R_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>为短半轴即地球极半径</p>
<img src="http://cos.pansis.site/202309281509418.png/abc123" alt="image-20230928150943328" style="zoom:33%;" />
<p>2、适用范围：研究惯性导航问题时</p>
<p>3、椭圆度（扁率）</p>
<img src="http://cos.pansis.site/202309281510745.png/abc123" alt="image-20230928151032694" style="zoom: 50%;" />
<p>4、重力方向：旋转椭球面的法线方向来代替重力方向</p>
<h4 id="3-大地水准体">3、大地水准体</h4>
<p>1、定义：把平均海平面延伸穿过所有陆地地块，则所形成的几何体称为大地水准体。</p>
<p>2、旋转椭球体与大 地水准体基本相符。</p>
<h4 id="4-地球参考椭球的曲率半径">4、地球参考椭球的曲率半径</h4>
<p>1、在地球表面不同的地点其曲率半径也不相同。</p>
<p>2、子午圈：过极轴和P点的平面与椭球平面的交线</p>
<p>​      卯酉圈：过P点的法线且垂直于过P点的子午面的平面与椭球表面的交线</p>
<p>3、同一地点，<strong>子午圈曲率半径</strong>与<strong>卯酉圈曲率半径</strong>也不相同。</p>
<img src="http://cos.pansis.site/202309281540773.png/abc123" alt="image-20230928152124701" style="zoom: 50%;" />
<p>4、曲率半径公式</p>
<p>子午圈曲率半径 <img src="http://cos.pansis.site/202310130753001.png/abc123" alt="image-20231013075332848" style="zoom:33%;" /></p>
<p>卯酉圈曲率半径  <img src="http://cos.pansis.site/202310130753067.png/abc123" alt="image-20231013075348989" style="zoom:33%;" /></p>
<h2 id="二-经纬度">二、经纬度</h2>
<h4 id="1-纬度">1、纬度</h4>
<p>1、地心纬度：地球上一点与地球几何中心的连线和地球赤道平面之间的夹角</p>
<p>主要用于导航问题。</p>
<p>2、地理纬度：地球子午圈上某点 的法线与地球赤道平面的夹角。</p>
<p>主要用于大地测量。</p>
<img src="http://cos.pansis.site/202309281539944.png/abc123" alt="image-20230928152501201" style="zoom:33%;" />
<h4 id="2-经度">2、经度</h4>
<p>1、定义：通过地球上某一点的本地子午面与本初子午面（0度经线）的夹角。</p>
<img src="http://cos.pansis.site/202309281539344.png/abc123" alt="image-20230928152528912" style="zoom:33%;" />
<h2 id="三-地球重力场">三、地球重力场</h2>
<h4 id="1-地球重力场组成">1、地球重力场组成</h4>
<p>1、在非惯性坐标系看来，地球重力场由地球引力场和地球自转的离心惯性力形成（两者合成重力场，形成重力加速度）。</p>
<p>2、地球引力：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><msub><mi>G</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">mG_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathdefault">m</span><span class="mord"><span class="mord mathdefault">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 指向地心</p>
<p>3、离心惯性力：$-m\overrightarrow{w_{ie}} × (\overrightarrow{w_{ie}}×\overrightarrow{r}) $</p>
<h4 id="2-重力加速度">2、重力加速度</h4>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>g</mi><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.147em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9525600000000001em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span><span class="svg-align" style="top:-3.43056em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span></span></span></p>
<p>1、重力加速度只在赤道和两极指向地心。</p>
<p>2、数值大小与所在的地理纬度L和高度h有关</p>
<p>​    <strong>计算公式</strong>  <img src="http://cos.pansis.site/202310130754116.png/abc123" alt="image-20231013075434952" style="zoom:33%;" /></p>
<h4 id="3-地球重力场特性">3、地球重力场特性</h4>
<p>1、重力异常：实际重力加速度与理论重力加速度在数值上的偏差称为重力异常。</p>
<p>原因：地球并非理想的旋转椭球体，地球内部局部地区密度不均</p>
<p>2、垂线偏差：由于地球自转影响，引力加速度与重力加速度在数值和方向上存在着差异</p>
<p>数值差异公式  <img src="http://cos.pansis.site/202310130754154.png/abc123" alt="image-20231013075456048" style="zoom:33%;" /></p>
<p>方向差异公式  <img src="http://cos.pansis.site/202310130755735.png/abc123" alt="image-20231013075519630" style="zoom:33%;" /></p>
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